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===Theory of categories===

===Theory of categories===

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Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers:  "Category" from Aristotle and Kant, "Functor" from Carnap (''Logische Syntax der Sprache''), and "natural transformation" from then current informal parlance. ([[Saunders Mac Lane]], ''Categories for the Working Mathematician'', 29–30).

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<p>Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers:  "Category" from Aristotle and Kant, "Functor" from Carnap (''Logische Syntax der Sprache''), and "natural transformation" from then current informal parlance.</p>

Mac Lane did not mention Peirce among the objects of his sincerest flattery, but he might as well have, for his mention of Aristotle and Kant well enough credits his deep indebtedness to the pursers of them all.  As [[Richard Feynman]] was fond of observing, 'the same questions have the same answers', and the problem that a system of categories is aimed to 'beautify' is the same sort of beast whether it's Aristotle, Kant, Peirce, Carnap, or [[Eilenberg]] and Mac Lane that bends the bow. What is that problem?  To answer that, let's begin again at the source:

<p>(Saunders Mac&nbsp;Lane, ''Categories for the Working Mathematician'', 29&ndash;30).</p>

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<p>Things are equivocally named, when they have the name only in common, the definition (or statement of essence) corresponding with the name being different.  For instance, while a man and a portrait can properly both be called ''animals'' (&#950;&#969;&#959;&#957;), these are equivocally named.  For they have the name only in common, the definitions (or statements of essence) corresponding with the name being different.  For if you are asked to define what the being an animal means in the case of the man and the portrait, you give in either case a definition appropriate to that case alone.</p>

<p>Things are equivocally named, when they have the name only in common, the definition (or statement of essence) corresponding with the name being different.  For instance, while a man and a portrait can properly both be called ''animals'' (&#950;&#969;&#959;&#957;), these are equivocally named.  For they have the name only in common, the definitions (or statements of essence) corresponding with the name being different.  For if you are asked to define what the being an animal means in the case of the man and the portrait, you give in either case a definition appropriate to that case alone.</p>

<p>Things are univocally named, when not only they bear the same name but the name means the same in each case — has the same definition corresponding.  Thus a man and an ox are called ''animals''.  The name is the same in both cases;  so also the statement of essence.  For if you are asked what is meant by their both of them being called ''animals'', you give that particular name in both cases the same definition. (Aristotle, ''Categories'', 1.1^a1–12).</p>

<p>Things are univocally named, when not only they bear the same name but the name means the same in each case — has the same definition corresponding.  Thus a man and an ox are called ''animals''.  The name is the same in both cases;  so also the statement of essence.  For if you are asked what is meant by their both of them being called ''animals'', you give that particular name in both cases the same definition.</p>

 

<p>(Aristotle, ''Categories'', 1.1^a1–12).</p>

In the logic of Aristotle categories are adjuncts to reasoning that are designed to resolve equivocations and thus to prepare  ambiguous signs, that are otherwise recalcitrant to being ruled by logic, for the application of logical laws.  An equivocation is a variation in meaning, or a manifold of sign senses, and so Peirce's claim that three categories are sufficient amounts to an assertion that all manifolds of meaning can be unified in just three steps.

In the logic of Aristotle categories are adjuncts to reasoning that are designed to resolve equivocations and thus to prepare  ambiguous signs, that are otherwise recalcitrant to being ruled by logic, for the application of logical laws.  An equivocation is a variation in meaning, or a manifold of sign senses, and so Peirce's claim that three categories are sufficient amounts to an assertion that all manifolds of meaning can be unified in just three steps.

The following passage is critical to the understanding of Peirce's Categories:

The following passage is critical to the understanding of Peirce's Categories:

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<p>I will now say a few words about what you have called Categories, but for which I prefer the designation Predicaments, and which you have explained as predicates of predicates.</p>

<p>I will now say a few words about what you have called Categories, but for which I prefer the designation Predicaments, and which you have explained as predicates of predicates.</p>

<p>My thoughts on this subject are not yet harvested.  I will only say that the subject concerns Logic, but that the divisions so obtained must not be confounded with the different Modes of Being:  Actuality, Possibility, Destiny (or Freedom from Destiny).</p>

<p>My thoughts on this subject are not yet harvested.  I will only say that the subject concerns Logic, but that the divisions so obtained must not be confounded with the different Modes of Being:  Actuality, Possibility, Destiny (or Freedom from Destiny).</p>

<p>On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being.  Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into ''Realms'' for the different Predicaments. (Peirce, CP 4.549, "Prolegomena to an Apology for Pragmaticism", ''The Monist'' 16, 492–546 (1906), CP 4.530–572).</p>

<p>On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being.  Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into ''Realms'' for the different Predicaments.</p>

 

<p>(Peirce, CP 4.549, "Prolegomena to an Apology for Pragmaticism", ''The Monist'' 16, 492&ndash;546 (1906), CP 4.530&ndash;572).</p>

The first thing to extract from this passage is the fact that Peirce's Categories, or "Predicaments", are predicates of predicates.  Meaningful predicates have both ''[[extension (semantics)|extension]]'' and ''[[intension]]'', so predicates of predicates get their meanings from at least two sources of information, namely, the classes of relations and the qualities of qualities to which they refer.  Considerations like these tend to generate hierarchies of subject matters, extending through what is traditionally called the ''logic of second intensions'', or what is handled very roughly by ''[[second order logic]]'' in contemporary parlance, and continuing onward through ''higher intensions'', or ''[[higher order logic]]'' and ''[[type theory]]''.

The first thing to extract from this passage is the fact that Peirce's Categories, or "Predicaments", are predicates of predicates.  Meaningful predicates have both ''[[extension (semantics)|extension]]'' and ''[[intension]]'', so predicates of predicates get their meanings from at least two sources of information, namely, the classes of relations and the qualities of qualities to which they refer.  Considerations like these tend to generate hierarchies of subject matters, extending through what is traditionally called the ''logic of second intensions'', or what is handled very roughly by ''[[second order logic]]'' in contemporary parlance, and continuing onward through ''higher intensions'', or ''[[higher order logic]]'' and ''[[type theory]]''.